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Ultimate Limit-State Design using none toconnect none for web,windows applicationmake pdf-417 c# xav GS1 DataBar Overview xav = 1 x dy b 0 xmin xav = 1 x dy b 0 xav xmax Figure 6.9. A schemat none for none ic of membrane stress distributions inside the plate under predominantly longitudinal compressive loads: (a) before buckling; (b) after buckling, unloaded edges move freely in plane; and (c) after buckling, unloaded edges remain straight.

. xmax xmin xav 1 b = x dy b 0 xav ymin ymax equations, with regar none for none d to the unknown amplitudes Amn , will be obtained. The nonlinear stress distribution inside the plate can then be obtained from Eq. (6.

13) with Amn . Figure 6.9 shows a typical example of the axial membrane stress distribution inside a plate under predominantly longitudinal compressive loading before and after buckling occurs.

It is important to realize that the membrane stress distribution in the loading x direction can become nonuniform as the plate de ects from many causes, including buckling, initial de ection, and lateral pressure loading. The membrane stress distribution in the y direction also becomes nonuniform as long as the unloaded plate edges remain straight. However, no membrane stresses will develop in the y direction if the unloaded plate edges move freely in plane as long as no axial loading is applied in the y direction.

As is apparent from Figure 6.9, the maximum compressive membrane stresses, x max and y max , are developed around the plate edges that remain straight, but the minimum membrane stresses, x min and y min , occur in the middle of the plate where a membrane tension eld is formed by the plate de ection because the plate edges remain straight. The location of the maximum compressive stresses depends on the residual stresses.

If no residual stresses exist, the maximum compressive stresses will develop along the edges. However, when there are residual stresses, the maximum compressive stresses may be located inside the plate at the limits that are the tensile residual stress block breadths from the plate edges (Paik and Thayamballi 2003)..

6.5 Ultimate Strength of Plates T C Simply supported edges Simply supported edges xmax ymax x xmax ymin xmin C Simply supported edges Figure 6.10. Three po none for none ssible locations for the initial plastic yield at the plate edges under combined loads: (a) plasticity at corners; (b) plasticity at longitudinal edges; and (c) plasticity at transverse edges.

( ) expected yielding locations; T: tension; C: compression.. ymax With an increase in t none none he plate de ection, the upper and/or lower bers inside the middle of the plate will initially yield by bending. However, as long as it is possible to redistribute the applied loads to the straight plate boundaries by membrane action, the plate will not collapse. Collapse will occur when the most stressed boundary locations yield because the plate cannot keep the boundaries straight any further.

This results in a rapid increase of lateral plate de ection. Because of the nature of combined membrane axial stresses in the x and y directions, three possible locations for initial yield at edges are generally considered: plate corners, longitudinal edges, and transverse edges; see Figure 6.10.

The stress status for the two edge locations, that is, at each longitudinal or transverse edge, can be expected to be the same as long as the longitudinal or transverse axial stresses are uniformly applied, that is, without in-plane bending. Depending on the predominant half-wave mode in the long direction, the location of the possible plasticity can vary at the long edges because the location of the minimum membrane stresses can be different; however, it is always the mid-edges in the short direction. Yielding can be assessed by using the Mises Henckey yield criterion.

Therefore, the three resulting ultimate strength criteria for the most probable yield locations may be found as follows: (1) Yielding at corners: x max Y. x max Y y max Y y max Y av Y = 1.. (6.20a). Ultimate Limit-State Design (2) Yielding at longitudinal edges: x max Y x max Y y min Y y min Y av Y = 1.. (6.20b). (3) Yielding at transverse edges: x min Y x min Y y max Y y max Y av Y = 1.. (6.20c). Because the maximum o none none r minimum membrane stresses of plates are expressed as functions of applied stress components, as well as initial de ections and welding residual stresses, Eqs. (6.20a) (6.

20c) are nonlinear functions. The smallest value among the solutions of these functions with regard to applied stress components will become the plate ultimate strength. This theory has been added to ALPS/ULSAP (2006).

. 6.5.4 Semianalytical Methods In these methods, the geometrical nonlinearity-related behavior of plates that is, elastic large de ection behavior is analyzed by direct solutions of nonlinear governing differential equations and material nonlinearities (i.

e., plasticity) are evaluated by numerical techniques to account for the effect of progressive plasticity expansion with increase in the applied loads. Paik et al.

(2001) developed a semianalytical method using this concept. For this purpose, it is assumed that the added de ection wi and stress function Fi at the end of the ith step of load increment are calculated by wi = wi 1 + Fi = Fi 1 + w, F, (6.21a) (6.

21b). where w and F are the none none increments of de ection or stress functions, respectively, where the pre x indicates the increment of the variable. The incremental forms of governing differential equations of large de ection plate theory, Eqs. (6.

12a) and (6.12b), are derived as follows: 4 w 4 w 4 w +2 2 2 + x 4 x y y4 2 Fi 1 2 w 2 F 2 (wi 1 + w0 ) 2 Fi 1 2 w + + y2 x 2 y2 x 2 x 2 y2. 2 F 2 (wi 1 + wo ) p 2 F 2 (wi 1 + wo ) 2 Fi 1 2 w 2 + = 0, 2 x 2 y2 x y x y x y x y t (6.22a).
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